reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem Th63:
for R being add-associative right_zeroed right_complementable
            Abelian non empty doubleLoopStr,
    a being Element of R,
    i,j being Integer holds (i - j) '*' a = i '*' a - j '*' a
proof
let R be add-associative right_zeroed right_complementable
         Abelian non empty doubleLoopStr,
    a be Element of R, i,j be Integer;
thus (i - j) '*' a
    = (i '*' a) + ((0-j) '*' a) by Th61
   .= i '*' a - j '*' a by Th62;
end;
